In mathematics, a square root of a number a is a number y such that y2 = a, in other words, a number y whose square (the result of multiplying the number by itself, or y × y) is a.[1] For example, 4 and −4 are square roots of 16 because 42 = (−4)2 = 16.

Every non-negative real numbera has a unique non-negative square root, called the principal square root, which is denoted by √a, where √ is called the radical sign or radix. For example, the principal square root of 9 is 3, denoted √9 = 3, because 32 = 3 × 3 = 9 and 3 is non-negative. The term whose root is being considered is known as the radicand. The radicand is the number or expression underneath the radical sign, in this example 9.

Every positive number a has two square roots: √a, which is positive, and −√a, which is negative. Together, these two roots are denoted ± √a (see ± shorthand). Although the principal square root of a positive number is only one of its two square roots, the designation "the square root" is often used to refer to the principal square root. For positive a, the principal square root can also be written in exponent notation, as a1/2.[2]

Square roots of negative numbers can be discussed within the framework of complex numbers. More generally, square roots can be considered in any context in which a notion of "squaring" of some mathematical objects is defined (including algebras of matrices, endomorphism rings, etc.)

In Ancient India, the knowledge of theoretical and applied aspects of square and square root was at least as old as the Sulba Sutras, dated around 800–500 BC (possibly much earlier).[citation needed] A method for finding very good approximations to the square roots of 2 and 3 are given in the Baudhayana Sulba Sutra.[5]Aryabhata in the Aryabhatiya (section 2.4), has given a method for finding the square root of numbers having many digits.

In the Chinese mathematical work Writings on Reckoning, written between 202 BC and 186 BC during the early Han Dynasty, the square root is approximated by using an "excess and deficiency" method, which says to "...combine the excess and deficiency as the divisor; (taking) the deficiency numerator multiplied by the excess denominator and the excess numerator times the deficiency denominator, combine them as the dividend."[7]

Mahāvīra, a 9th-century Indian mathematician, was the first to state that square roots of negative numbers do not exist.[8]

The graph of the function f(x) = √x, made up of half a parabola with a vertical directrix.

The principal square root function f(x) = √x (usually just referred to as the "square root function") is a function that maps the set of non-negative real numbers onto itself. In geometrical terms, the square root function maps the area of a square to its side length.

By trial-and-error, one can square an estimate for √a and raise or lower the estimate until it agrees to sufficient accuracy. For this technique it's prudent to use the identity

as it allows one to adjust the estimate x by some amount c and measure the square of the adjustment in terms of the original estimate and its square. Furthermore, when c is close to 0, because the tangent line to the graph of at c=0, as a function of c alone, is . Thus, small adjustments to x can be planned out by setting to , or .

The most common iterative method of square root calculation by hand is known as the "Babylonian method" or "Heron's method" after the first-century Greek philosopher Heron of Alexandria, who first described it.[14] The method uses the same iterative scheme as the Newton–Raphson method yields when applied to the function y = f(x)=x2 − a, using the fact that its slope at any point is but predates it by many centuries.[15] The algorithm is to repeat a simple calculation that results in a number closer to the actual square root each time it is repeated with its result as the new input. The motivation is that if x is an overestimate to the square root of a non-negative real number a then a/x will be an underestimate and so the average of these two numbers is a better approximation than either of them. However, the inequality of arithmetic and geometric means shows this average is always an overestimate of the square root (as noted below), and so it can serve as a new overestimate with which to repeat the process, which converges as a consequence of the successive overestimates and underestimates being closer to each other after each iteration. To find x :

Start with an arbitrary positive start value x. The closer to the square root of a, the fewer the iterations that will be needed to achieve the desired precision.

Replace x by the average (x + a/x) / 2 between x and a/x.

Repeat from step 2, using this average as the new value of x.

That is, if an arbitrary guess for √a is , and xn+1 = (xn + a/xn)/2, then each xn is an approximation of √a which is better for large n than for small n. If a is positive, the convergence is quadratic, which means that in approaching the limit, the number of correct digits roughly doubles in each next iteration. If a = 0, the convergence is only linear.

Using the identity

the computation of the square root of a positive number can be reduced to that of a number in the range [1,4). This simplifies finding a start value for the iterative method that is close to the square root, for which a polynomial or piecewise-linearapproximation can be used.

The time complexity for computing a square root with n digits of precision is equivalent to that of multiplying two n-digit numbers.

Using the Riemann surface of the square root, it is shown how the two leaves fit together

The square of any positive or negative number is positive, and the square of 0 is 0. Therefore, no negative number can have a real square root. However, it is possible to work with a more inclusive set of numbers, called the complex numbers, that does contain solutions to the square root of a negative number. This is done by introducing a new number, denoted by i (sometimes j, especially in the context of electricity where "i" traditionally represents electric current) and called the imaginary unit, which is defined such that i2 = −1. Using this notation, we can think of i as the square root of −1, but notice that we also have (−i)2 = i2 = −1 and so −i is also a square root of −1. By convention, the principal square root of −1 is i, or more generally, if x is any non-negative number, then the principal square root of −x is

The right side (as well as its negative) is indeed a square root of −x, since

For every non-zero complex number z there exist precisely two numbers w such that w2 = z: the principal square root of z (defined below), and its negative.

To find a definition for the square root that allows us to consistently choose a single value, called the principal value, we start by observing that any complex number x + iy can be viewed as a point in the plane, (x, y), expressed using Cartesian coordinates. The same point may be reinterpreted using polar coordinates as the pair (r, φ), where r ≥ 0 is the distance of the point from the origin, and φ is the angle that the line from the origin to the point makes with the positive real (x) axis. In complex analysis, this value is conventionally written reiφ. If

then we define the principal square root of z as follows:

The principal square root function is thus defined using the nonpositive real axis as a branch cut. The principal square root function is holomorphic everywhere except on the set of non-positive real numbers (on strictly negative reals it isn't even continuous). The above Taylor series for √1 + x remains valid for complex numbers x with |x| < 1.

When the number is expressed using Cartesian coordinates the following formula can be used for the principal square root:[16][17]

where sgn is the signum function. The sign of the imaginary part of the root is taken to be the same as the sign of the imaginary part of the original number. The real part of the principal value is always non-negative.

As the other square root is simply −1 times the principal square root, both roots can be written as

Because of the discontinuous nature of the square root function in the complex plane, the law √zw = √z√w is in general not true. (Equivalently, the problem occurs because of the freedom in the choice of branch. The chosen branch may or may not yield the equality; in fact, the choice of branch for the square root need not contain the value of √z√w at all, leading to the equality's failure. A similar problem appears with the complex logarithm and the relation log z + log w = log(zw).) Wrongly assuming this law underlies several faulty "proofs", for instance the following one showing that −1 = 1:

The third equality cannot be justified (see invalid proof). It can be made to hold by changing the meaning of √ so that this no longer represents the principal square root (see above) but selects a branch for the square root that contains (√−1)·(√−1). The left-hand side becomes either

if the branch includes +i or

if the branch includes −i, while the right-hand side becomes

where the last equality, √1 = −1, is a consequence of the choice of branch in the redefinition of √.

If A is a positive-definite matrix or operator, then there exists precisely one positive definite matrix or operator B with B2 = A; we then define A1/2 = √A = B. In general matrices may have multiple square roots or even an infinitude of them. For example the 2 × 2identity matrix has an infinity of square roots.[18]

Each element of an integral domain has no more than 2 square roots. The difference of two squares identity u2 − v2 = (u − v)(u + v) is proved using the commutativity of multiplication. If u and v are square roots of the same element, then u2 − v2 = 0. Because there are no zero divisors this implies u = v or u + v = 0, where the latter means that two roots are additive inverses of each other. In other words, the square root of an element, if it exists, is unique up to a sign. The only square root of 0 in an integral domain is 0 itself.

In a field of characteristic 2, an element has either one square root, because each element is its own additive inverse, or does not have any at all (if the field is finite of characteristic 2 then every element has a unique square root). In a field of any other characteristic, any non-zero element either has two square roots, as explained above, or does not have any.

Given an odd prime numberp, let q = pe for some positive integer e. A non-zero element of the field Fq with q elements is a quadratic residue if it is has a square root in Fq. Otherwise, it is a quadratic non-residue. There are (q − 1)/2 quadratic residues and (q − 1)/2 quadratic non-residues; zero is not counted in either class. The quadratic residues form a group under multiplication. The properties of quadratic residues are widely used in number theory.

In a ring we call an element b a square root of aiffb2 = a. To see that the square root need not be unique up to sign in a general ring, consider the ring from modular arithmetic. Here, the element 1 has four distinct square roots, namely ±1 and ±3. On the other hand, the element 2 has no square root. See also the article quadratic residue for details.

The square root of 0 is by definition either 0 or a zero divisor, and where zero divisors do not exist (such as in quaternions and, generally, in division algebras), it is uniquely 0. It is not necessarily true in general rings, where Z/n2Z for any natural n provides an easy counterexample.

One of the most intriguing results from the study of irrational numbers as continued fractions was obtained by Joseph Louis Lagrangec. 1780. Lagrange found that the representation of the square root of any non-square positive integer as a continued fraction is periodic. That is, a certain pattern of partial denominators repeats indefinitely in the continued fraction. In a sense these square roots are the very simplest irrational numbers, because they can be represented with a simple repeating pattern of integers.

√2

= [1; 2, 2, ...]

√3

= [1; 1, 2, 1, 2, ...]

√4

= [2]

√5

= [2; 4, 4, ...]

√6

= [2; 2, 4, 2, 4, ...]

√7

= [2; 1, 1, 1, 4, 1, 1, 1, 4, ...]

√8

= [2; 1, 4, 1, 4, ...]

√9

= [3]

√10

= [3; 6, 6, ...]

√11

= [3; 3, 6, 3, 6, ...]

√12

= [3; 2, 6, 2, 6, ...]

√13

= [3; 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, ...]

√14

= [3; 1, 2, 1, 6, 1, 2, 1, 6, ...]

√15

= [3; 1, 6, 1, 6, ...]

√16

= [4]

√17

= [4; 8, 8, ...]

√18

= [4; 4, 8, 4, 8, ...]

√19

= [4; 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, ...]

√20

= [4; 2, 8, 2, 8, ...]

The square bracket notation used above is a sort of mathematical shorthand to conserve space. Written in more traditional notation the simple continued fraction for the square root of 11, [3; 3, 6, 3, 6, ...], looks like this:

where the two-digit pattern {3, 6} repeats over and over again in the partial denominators. Since 11 = 32 + 2, the above is also identical to the following generalized continued fractions:

The square root of a positive number is usually defined as the side length of a square with the area equal to the given number. But the square shape is not necessary for it: if one of two similarplanar Euclidean objects has the area a times greater than another, then the ratio of their linear sizes is √a.

The construction is also given by Descartes in his La Géométrie, see figure 2 on page 2. However, Descartes made no claim to originality and his audience would have been quite familiar with Euclid.

Euclid's second proof in Book VI depends on the theory of similar triangles. Let AHB be a line segment of length a + b with AH = a and HB = b. Construct the circle with AB as diameter and let C be one of the two intersections of the perpendicular chord at H with the circle and denote the length CH as h. Then, using Thales' theorem and, as in the proof of Pythagoras' theorem by similar triangles, triangle AHC is similar to triangle CHB (as indeed both are to triangle ACB, though we don't need that, but it is the essence of the proof of Pythagoras' theorem) so that AH:CH is as HC:HB, i.e. from which we conclude by cross-multiplication that and finally that . Note further that if you were to mark the midpoint O of the line segment AB and draw the radius OC of length then clearly OC > CH, i.e. (with equality if and only if a = b), which is the arithmetic–geometric mean inequality for two variables and, as noted above, is the basis of the Ancient Greek understanding of "Heron's method".

Another method of geometric construction uses right triangles and induction: √1 can, of course, be constructed, and once √x has been constructed, the right triangle with 1 and √x for its legs has a hypotenuse of √x + 1. The Spiral of Theodorus is constructed using successive square roots in this manner.